Re: Trig plotting and direct substitution

Both of your questions involve taking the first and second derivative of your function.

In (i), the first derivative is useful for finding critical points and seeing where the function is increasing or decreasing. The second derivative is used to see where the function is concave up or concave down. Your teacher should have reviewed a list of steps for drawing graphs this way.

In (ii), it is as it says, direct substitution. d^2s/dt^2 is the second derivative of s, ds/dt is the first derivative of s, and s is s. Plug in on the left hand side and see if it equals 0

Re: Trig plotting and direct substitution

I'm not sure if you need the formula y = a sin(bx+c) + d at all, but it should be noted that if you do know what the base function looks like, you can apply a transformation on that graph to get the desired graph of a particular function.

For example, if you know the amplitude, period, etc, of sin(x), we can derive the amplitude, period, etc, of a sin(bx+c) + d using the theory of transformations, by noting that
to transform sin(x) -> a sin(bx+c) + d we have

1) A vertical stretch by a factor of a
2) A horizontal stretch by a factor of 1/b
3) A vertical displacement by a factor of d
4) A horizontal displacement by a factor of -c/b